CHAPTER 1 Derivation of Telegrapher's Equations and Field-To

CHAPTER 1

Derivation of telegrapher’s equations and ﬁ eld-to-transmission line interaction

C.A. Nucci1, F. Rachidi2 & M. Rubinstein3 1University of Bologna, Bologna, Italy. 2Swiss Federal Institute of Technology, Lausanne, Switzerland. 3Western University of Applied Sciences, Yverdon, Switzerland.

Abstract

In this chapter, we discuss the transmission line theory and its application to the problem of external electromagnetic fi eld coupling to transmission lines. After a short discussion on the underlying assumptions of the transmission line theory, we start with the derivation of fi eld-to-transmission line coupling equations for the case of a single wire line above a perfectly conducting ground. We also describe three seemingly different but completely equivalent approaches that have been proposed to describe the coupling of electromagnetic fi elds to transmission lines. The derived equations are extended to deal with the presence of losses and multiple conductors. The time-domain representation of fi eld-to-transmission line coupling equations, which allows a straightforward treatment of non-linear phenomena as well as the variation in the line topology, is also described. Finally, solution methods in frequency domain and time domain are presented.

1 Transmission line approximation

The problem of an external electromagnetic ﬁ eld coupling to an overhead line can be solved using a number of approaches. One such approach makes use of antenna theory, a general methodology based on Maxwell’s equations [1]. Differ- ent methods based on this approach generally use the thin-wire approximation, in which the wire’s cross section is assumed to be much smaller than the minimum signiﬁ cant wavelength. When electrically long lines are involved, however, the antenna theory approach requires prohibitively long computational times and high computer resources. On the

WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-84564-063-7/01 4 Electromagnetic Field Interaction with Transmission Lines other hand, the less resource hungry quasi-static approximation [1], in which propagation is neglected and coupling is described by means of lumped elements, can be adopted only when the overall dimensions of the circuit are smaller than the minimum signifi cant wavelength of the electromagnetic fi eld. For many practical cases, however, this condition is not satisfi ed. As an example, let us consider the case of power lines illuminated by a lightning electromagnetic pulse (LEMP). Power networks extend, in general, over distances of several kilometers, much larger than the minimum wavelengths associated with LEMP. Indeed, signifi cant portions of the frequency spectrum of LEMP extend to frequencies up to a few MHz and beyond, which corresponds to minimum wavelengths of about 100 m or less [2]. A third approach is known as transmission line theory. The main assumptions for this approach are as follows: 1. Propagation occurs along the line axis. 2. The sum of the line currents at any cross section of the line is zero. In other words, the ground – the reference conductor – is the return path for the currents in the n overhead conductors. 3. The response of the line to the coupled electromagnetic fi elds is quasi-transverse electromagnetic (quasi-TEM) or, in other words, the electromagnetic fi eld produced by the electric charges and currents along the line is confi ned in the transverse plane and perpendicular to the line axis. If the cross-sectional dimensions of the line are electrically small, propagation can indeed be assumed to occur essentially along the line axis only and the fi rst assumption can be considered to be a good approximation. The second condition is satisfi ed if the ground plane exhibits infi nite conductivity since, in that case, the currents and voltages can be obtained making use of the method of images, which guarantees currents of equal amplitude and opposite direction in the ground. The condition that the response of the line is quasi-TEM is satisfi ed only up to a threshold frequency above which higher-order modes begin to appear [1]. For some cases, such as infi nite parallel plates or coaxial lines, it is possible to derive an exact expression for the cutoff frequency below which only the TEM mode exists [3]. For other line structures (i.e. multiple conductors above a ground plane), the TEM mode response is generally satisfi ed as long as the line cross section is electrically small [3]. Under these conditions, the line can be represented by a distributed-parameter structure along its axis. For uniform transmission lines with electrically small cross-sectional dimensions (not exceeding about one-tenth of the minimum signifi cant wavelength of the exciting electromagnetic fi eld), a number of theoretical and experimental studies have shown a fairly good agreement between results obtained using the transmission line approximation and results obtained either by means of antenna theory or experiments [4]. A detailed discussion of the validity of the basic assumptions of the transmission line theory is beyond the scope of this chapter. However, it is worth noting that, by assuming that the sum of all the currents is equal to zero, we are

WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Derivation of Telegrapher’s Equations 5 considering only ‘transmission line mode’ currents and neglecting the so-called ‘antenna-mode’ currents [1]. If we wish to compute the load responses of the line, this assumption is adequate, because the antenna mode current response is small near the ends of the line. Along the line, however, and even for electrically small line cross sections, the presence of antenna-mode currents implies that the sum of the currents at a cross section is not necessarily equal to zero [1, 3]. However, the quasi-symmetry due to the presence of the ground plane results in a very small contribution of antenna mode currents and, consequently, the predominant mode on the line will be transmission line [1].

2 Single-wire line above a perfectly conducting ground

We will consider fi rst the case of a lossless single-wire line above a perfectly conducting ground. This simple case will allow us to introduce various coupling models and to discuss a number of concepts essential to the understanding of the electromagnetic fi eld coupling phenomenon. Later in this chapter (Sections 4 and 5), we will cover the cases of lossy and multiconductor lines. The transmission line is defi ned by its geometrical parameters (wire radius a and height above ground h) and its terminations ZA and ZB, as illustrated in Fig. 1, where the line is illuminated by an external electromagnetic fi eld. The problem of interest is the calculation of the induced voltages and currents along the line and at the terminations. e e The external exciting electric and magnetic fi elds E and B are defi ned as the i i r r sum of the incident fi elds, E and B , and the ground-refl ected fi elds, E and B , determined in absence of the line conductor. The total fi elds E and B at a given point in space are given by the sum of the excitation fi elds and the scattered fi elds from the line, the latter being denoted as Es and Bs. The scattered fi elds are created by the currents and charges fl owing in the line conductor and by the corresponding currents and charges induced in the ground. Three seemingly different but completely equivalent approaches have been proposed to describe the coupling of electromagnetic fi elds to transmission lines. In what

z E e

e B 2a

y C h ZA ZB

x 0Lx x + dx Figure 1: Geometry of the problem.

WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 6 Electromagnetic Field Interaction with Transmission Lines follows, we will present each one of them in turn. We will ﬁ rst derive the ﬁ eld-to- transmission line coupling equations (which are sometimes referred to as generalized telegrapher’s equations) following the development of Taylor et al. [5].

2.1 Taylor, Satterwhite and Harrison model

2.1.1 Derivation of the ﬁ rst ﬁ eld-to-transmission line coupling (generalized telegrapher’s) equation Consider the single conductor transmission line of height h in Fig. 1. Applying Stokes’ theorem to Maxwell’s equation ∇E = –jwB for the area enclosed by the closed contour C yields ⋅=− ⋅ ∫∫∫Eldd jw BeSy (1) C S Since the contour has a differential width ∆x, eqn (1) can be written as

hxx+∆ +∆ − + − ∫∫[Exzz ( xzExzz ,) (,)]d [ ExhEx xx (,) (,0)]d x 0 x h xx+∆ =− jBxzxzw∫∫ y (,)dd (2) 0 x (The coordinate y will be implicitly assumed to be 0 and, for the sake of clar- ity, we will omit the y-dependence unless the explicit inclusion is important for the discussion.) Dividing by ∆x and taking the limit as ∆x approaches zero yields ∂ hh ∫∫ExzzExhEx(,)d+−=− (,) (,0) jw Bxzz (,)d (3) ∂x zxx y 00 Since the wire and the ground are assumed to be perfect conductors, the total tangential electric ﬁ elds at their respective surfaces, Ex(x,h) and Ex(x,0), are zero. Deﬁ ning also the total transverse voltage V(x) in the quasi-static sense (since h << λ) as

h =− Vx()∫ Ez (,)d xz z (4) 0 Equation (3) becomes

d()Vx hhh =−jBxzzjBxzzjBxzzwww∫∫∫(,)d =−es (,)d − (,)d (5) dx yyy 000 where we have decomposed the B-ﬁ eld into the excitation and scattered components.

The last integral in eqn (5) represents the magnetic fl ux between the conductor and the ground produced by the current I(x) fl owing in the conductor. Now, Ampère–Maxwell’s equation in integral form is given by ∫∫∫BlIjs⋅=+ddw Ds ⋅ (6) C′ If we defi ne a path C' in the transverse plane defi ned by a constant x in such a manner that the conductor goes through it, eqn (6) can be rewritten as s ⋅= + ⋅ ∫∫∫BT (,,)d xyz l Ix () jw Dxx (,,) xyz a d s (7) C′ where the subscript T is used to indicate that the fi eld is in the transverse direction, ax is the unit vector in the x direction, and where we have explicitly included the dependence of the fi elds on the three Cartesian coordinates. If the response of the wire is TEM, the electric fl ux density D in the x direction is zero and eqn (7) can be written as s ⋅=l ∫ BxyzT (,,)d Ix () (8) C′ s

Clearly, I(x) is the only source of BT ( x,y,z). Further, it is apparent from eqn (8) s that BT ( x,y,z) is directly proportional to I(x). Indeed, if I(x) is multiplied by a con- s stant multiplicative factor which, in general, can be complex, B T( x) too will be multiplied by that factor. Further, the proportionality factor for a uniform cross- section line must be independent of x. s

Let us now concentrate on the y component of BT (x,y,z) for points in the plane s y = 0. Using the facts we just established that I(x) and B T (x) are proportional and that the proportionality factor is independent of x, we can now write

Bxys ( ,=== 0, z ) ky ( 0, zIx ) ( ) (9) y where k(y, z) is the proportionality constant. With this result, we now go back to the last integral in eqn (5),

h s ∫ Bxzzy (,)d 0 Note that, although the value of y is not explicitly given, y = 0. The integral represents the per-unit-length magnetic ﬂ ux under the line. Substituting eqn (9) into it, we obtain

h h s == ∫∫Bxzzy (,)d ky ( 0,)()d zIxz (10) 00

We can rewrite eqn (10) as follows

hh s == ∫∫BxzzIxkyy (,)d () ( 0,)d zz (11) 00 Equation (11) implies that the per-unit-length scattered magnetic ﬂ ux under the line at any point along it is proportional to the current at that point. The proportionality h ∫ constant, given by 0 k(y = 0,z)dz, is the per-unit-length inductance of the line L'. This results in the well-known linear relationship between the magnetic ﬂ ux and the line current:

h s = ′ ∫ Bxzzy (,)d LIx () (12) 0 Assuming that the radius of the wire is much smaller than the height of the line (a << h), the magnetic ﬂ ux density can be calculated using Ampere’s law and the ≅ π integral can be evaluated analytically, yielding L' (m0/2 )ln(2h/a) [1]. Inserting eqn (12) into eqn (5), we obtain the ﬁ rst generalized telegrapher’s equation

d()Vx h +=−jLIxww′ () j∫ Bxzze (,)d (13) dx y 0 Note that, unlike the classical telegrapher’s equations in which no external excitation is considered, the presence of an external fi eld results in a forcing function expressed in terms of the exciting magnetic fl ux. This forcing function can be viewed as a distributed voltage source along the line. Attention must be paid to the fact that the voltage V(x) in eqn (13) depends on the integration path since it is obtained by integration of an electric fi eld whose curl is not necessarily zero (eqn (4)).

2.1.2 Derivation of the second ﬁ eld-to-transmission line coupling equation To derive the second telegrapher’s equation, we will assume that the medium sur- = rounding the line is air (e e0) and we will start from the second Maxwell’s equation, ∇ × = H J + jwe0E, also called Ampère–Maxwell’s equation. Rearranging the terms and writing it in Cartesian coordinates for the z-component:

∂Bxz(,) Bxz J =−−1 y ∂ x (,) z jExzw z (,) (14) em∂∂xy e 00 0 σ The current density can be related to the E-ﬁ eld using Ohm’s law, J = air E, σ where air is the air conductivity. Since the air conductivity is generally low, we

WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Derivation of Telegrapher’s Equations 9 σ will assume here that air = 0 and will therefore neglect this term (which will eventually result in an equivalent parallel conductance in the coupling equation (see Section 5)). Integrating eqn (14) along the z axis from 0 to h, and making use of eqn (4), we obtain

h e e 1 ∂Bxz(,) ∂Bxz(,) −=jVxw ()∫ y −x d z em ∂∂xy 000 

h s s 1 ∂Bxz(,) ∂Bxz(,) +−y x dz (15) ∫ ∂∂ em00 xy 0 in which we have decomposed the magnetic fl ux density fi eld into the excitation and scattered components. Since the excitation fi elds are the fi elds that would exist if the line were not present, they must satisfy Maxwell’s equations. Applying Maxwell’s equation (14) to the components of the excitation electromagnetic fi eld and integrating along z from 0 to h directly under the line yields

hhe e 1 ∂B ∂B y −=x ddzjw Eze (16) ∫∫∂∂ z em00 xy 00 Using eqns (12) and (16) and given that the line response is assumed to be TEM, s Bx = 0, eqn (15) becomes

d(Ix ) h +=−jCVxww′′() jC∫ Ee (,)d xzz (17) dx z 0 where C' is the per-unit-length line capacitance related to the per-unit-length inductance through e0m0 = L'C'. Equation (17) is the second ﬁ eld-to-transmission line coupling equation. For a line of ﬁ nite length, such as the one represented in Fig. 1, the boundary conditions for the load currents and voltages must be enforced. They are simply given by VZI(0)=− (0) A (18)

VL()= ZIL () B (19)

2.1.3 Equivalent circuit Equations (13) and (17) are referred to as the Taylor model. They can be represented using an equivalent circuit, as shown in Fig. 2. The forcing functions (source

h ω∫ e j By(x,z)dz L'dx 0 I(x) I(x+dx)

h e ZA V(0) V(x)jω∫E (x,z)dz C'dx V(x+dx) V(L) ZB 0 z

0xx+dx L Figure 2: Equivalent circuit of a lossless single-wire overhead line excited by an electromagnetic ﬁ eld (Taylor et al. model). terms) in eqns (13) and (17) are included as a set of distributed series voltage and parallel current sources along the line.

2.2 Agrawal, Price and Gurbaxani model

An equivalent formulation of the fi eld-to-transmission line coupling equations was proposed in 1980 by Agrawal et al. [6]. This model is commonly referred to as the Agrawal model. We will call it the model of Agrawal et al. hereafter. The basis for the derivation of the Agrawal et al. model can be described as follows: The excitation fi elds produce a line response that is TEM. This response is expressed in terms of a scattered voltage V s(x), which is defi ned in terms of the line integral of the scattered electric fi eld from the ground to the line, and the total current I(x). The total voltage can be obtained from the scattered voltage through

h =+=−se s e VxVx() () Vx () Vx ()∫ Exzzz (,)d (20) 0 The ﬁ eld-to-transmission line coupling equations as derived by Agrawal et al. [6] are given by

s d()Vx+=′ e (21) jLIxw () Ex (,) xh dx d(Ix ) +=jCVxw ′ s () 0 (22) dx Note that, in this model, only one source term is present (in the ﬁ rst equation) and it is simply expressed in terms of the exciting electric ﬁ eld tangential to the line e conductor E x ( x,h).

e L'dx Ex(x,h)dx I(x)- + I(x+dx) h + + h e ∫ e ∫Ez(0,z)dz Ez(L,z)dz 0 S S S S 0 - V (0) V (x) C'dx V (x+dx) V (L) -

ZB ZA

0 x x+dx L Figure 3: Equivalent circuit of a lossless single-wire overhead line excited by an electromagnetic ﬁ eld (Agrawal et al. model).

The boundary conditions in terms of the scattered voltage and the total current as used in eqns (21) and (22), are given by

h se=− + VZIEzz(0)A (0)∫ z (0, )d (23) 0

h se=+ VL() ZILB ()∫ ELzzz (,)d (24) 0 The equivalent circuit representation of this model (eqns (21)–(24)) is shown in Fig. 3. For this model, the forcing function (the exciting electric fi eld tangential to the line conductor) is represented by distributed voltage sources along the line. In accordance with boundary conditions (23) and (24), two lumped voltage sources (equal to the line integral of the exciting vertical electric fi eld) are inserted at the line terminations. It is also interesting to note that this model involves only electric fi eld components of the exciting fi eld and the exciting magnetic fi eld does not appear explicitly as a source term in the coupling equations. As we will see in the next section where we present the Rachidi model [7], it is also possible to represent the coupling model in terms of magnetic fi elds only.

2.3 Rachidi model

Another form of the coupling equations, equivalent to the Agrawal et al. and to the Taylor et al. models, has been derived by Rachidi [7]. In this model, only the exciting magnetic ﬁ eld components appear explicitly as forcing functions in the equations:

d()Vx +=jLIxw ′ s () 0 (25) dx

dIs () x 1h ∂Bxze (,) +=jCVxw ′ ()∫ x d z (26) dx L′ ∂ y 0 in which I s(x) is the so-called scattered current related to the total current by

=+se Ix() I () x I () x (27) where the excitation current I e(x) is deﬁ ned as

1 h Ixee()=− ∫ Bxzz (,)d (28) L′ y 0 The boundary conditions corresponding to this formulation are

V(0) 1 h IBzzse(0)=− + ∫ (0, )d (29) ZL′ y A 0

VL() 1h ILse()=+∫ BLzz (,)d (30) ZL′ y B 0 The equivalent circuit corresponding to the above equivalent set of coupling equations is shown in Fig. 4. Note that the equivalent circuit associated with the Rachidi model could be seen as the dual circuit – in the sense of electrical net- work theory – of the one corresponding to the Agrawal et al. model (Fig. 3).

3 Contribution of the different electromagnetic ﬁ eld components

Nucci and Rachidi [8] have shown, on the basis of a speciﬁ c numerical example that, as predicted theoretically, the total induced voltage waveforms obtained using the three coupling models presented in Sections 2.1–2.3 are identical. However,

IS(x) L'dx IS(x+dx)

h h 1 e 1 e ∫B (0,z)dz ∫B (0,z)dz L' y h∂ e L' y 0 1 ∫ Bx V(L) 0 V(0) V(x) ∂ dz C'dx V(x+dx) L'0 y ZA ZB

0 x x+dx L Figure 4: Equivalent circuit of a lossless single-wire overhead line excited by an electromagnetic ﬁ eld (Rachidi model).

WIT Transactions on State of the Art in Science and Eng ineering, Vol 29, © 2008 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Derivation of Telegrapher’s Equations 13 the contribution of a given component of the exciting electromagnetic ﬁ eld to the total induced voltage and current varies depending on the adopted coupling model. Indeed, the three coupling models are different but fully equivalent approaches that predict identical results in terms of total voltages and total currents, in spite of the fact that they take into account the electromagnetic coupling in different ways. In other words, the three models are different expressions of the same equations, cast in terms of different combinations of the various electromagnetic ﬁ eld components, which are related through Maxwell’s equations.

4 Inclusion of losses

In the calculation of lightning-induced voltages, losses are, in principle, to be taken into account both in the wire and in the ground. Losses due to the ﬁ nite ground conductivity are the most important ones, and they affect both the electromagnetic ﬁ eld and the surge propagation along the line [9]. Let us make reference to the same geometry of Fig. 1, and let us now take into account losses both in the wire and in the ground plane. The wire conductivity and relative permittivity are sw and erw, respectively, and the ground, assumed to be homo- geneous, is characterized by its conductivity sg and its relative permittivity erg. The Agrawal et al. coupling equations extended to the present case of a wire above an imperfectly conducting ground can be written as (for a step by step derivation see [1])

s d()Vx+=′ e ZIx() Ex (,) xh (31) dx

d(Ix ) +=YV′ s () x 0 (32) dx where Z' and Y' are the longitudinal and transverse per-unit-length impedance and admittance respectively, given by [1, 9] (in [1], the per-unit-length transverse conductance has been disregarded) ZjLZZ′′′′=++w (33) wg ()GjCY′′′+ w Y ′ = g (34) GjCY′′′++w g in which

• L', C' and G' are the per-unit-length longitudinal inductance, transverse capacitance and transverse conductance, respectively, calculated for a lossless wire above a perfectly conducting ground:

mm− hh 2  Lha′ =≅00cosh1  ln   for >>(35) ππ   22aa

22ππee Cha′ =≅00for >> (36) −1 ln(2ha / ) cosh (ha / ) s GC′′= air (37) e 0

• Z'w is the per-unit-length internal impedance of the wire; assuming a round wire and an axial symmetry for the current, the following expression can be derived for the wire internal impedance [10]: ggI(a ) Z ′ = w0 w (38) w 2I()πaasg w1 w ______where gw = √ jwm0(sw + jwe0erw) is the propagation constant in the wire and I0 and I1 are the modifi ed Bessel functions of zero and fi rst order, respectively; • Z'g is the per-unit-length ground impedance, which is defi ned as [11, 12] h s jBxzxw ∫ y (,)d ′′=−−∞ ZjLg w (39) I s where B y is the y-component of the scattered magnetic induction fi eld. Several expressions for the ground impedance have been proposed in the literature ([13], see also Chapter 2). Sunde [14] derived a general expression for the ground impedance which is given by ∞ jwm −2hx ′ = 0 e Zxg ∫ d (40) π 22 0 xx++g g ______where gg = √ jwm0(sg + jwe0erg) is the propagation constant in the ground. As noted in [13], Sunde’s expression (40) is directly connected to the general expressions obtained from scattering theory. Indeed, it is shown in [1] that the general expression for the ground impedance derived using scattering theory reduces to the Sunde approximation when considering the transmission line approximation. Also, the results obtained using eqn (40) are shown to be accurate within the limits of the transmission line approximation [1]. The general expression (40) is not suitable for a numerical evaluation since it involves an integral over an infi nitely long interval. Several approximations for the ground impedance of a single-wire line have been proposed in the literature (see [11] for a survey). One of the simplest and most accurate was proposed by Sunde himself and is given by the following logarithmic function: + jwm 1 ggh Z ′ ≅ 0 ln  (41) g 2π g h g

It has been shown [11] that the above logarithmic expression represents an excellent approximation to the general expression (40) over the frequency range of interest.

Finally, Y'g is the so-called ground admittance, given by [1] g2 Y ′ ≅ g g ′ (42) Zg

5 Case of multiconductor lines

Making reference to the geometry of Fig. 5, the ﬁ eld-to-transmission line coupling equations for the case of a multi-wire system along the x-axis above an imperfectly conducting ground and in the presence of an external electromagnetic excitation are given by [1, 4, 15] d [Vxse ( )]++= jw [ L′′ ][ Ix ( )] Z [ Ix ( )] [ Exh ( , )] (43) iijiixigij dx

d ++′′ss = [Ixiijiiji ( )] [ G ][ V ( x )] jw [ C ][ V ( x )] [0] (44) dx in which s • [V i ( x)] and [Ii(x)] are frequency-domain vectors of the scattered voltage and the current along the line;

e E 2aj i

2ai k

2ak

h rik k

rij rjk

Ground Plane, sg, erg

Figure 5: Cross-sectional geometry of a multiconductor line in the presence of an external electromagnetic ﬁ eld.

e • [ E x ( x,hi)] is the vector of the exciting electric ﬁ eld tangential to the line conductors; • [0] is the zero-matrix (all elements are equal to zero);

• [L'ij] is the per-unit-length line inductance matrix. Assuming that the distances between conductors are much larger than their radii, the general expression for the mutual inductance between two conductors i and j is given by [1]

22 m rhh++() ′ = 0 ij i j Lij ln  (45) 2π rhh22+−() ij i j • The self-inductance for conductor i is given by  m0 2hi L′ = ln  (46) ii 2π r ii

• [C'ij] is the per-unit-length line capacitance matrix. It can be evaluated directly from the inductance matrix using the following expression [1] − []CL′′= em []1 (47) ij00 ij

• [G'ij] is the per-unit-length transverse conductance matrix. The transverse conductance matrix elements can be evaluated starting either from the capacitance matrix or the inductance matrix using the following relations:

s − []GC′′==air []sm [] L ′1 (48) ije ijair 0 ij 0

In most practical cases, the transverse conductance matrix elements G'ij are negligible in comparison with jwC'ij [3] and can therefore be neglected in the computation. • Finally, [Z' ] is the ground impedance matrix. The general expression for the gij mutual ground impedance between two conductors i and j derived by Sunde is given by [14]

∞ −+()hhx jwm ij ′ = 0 e Zrxxg ∫ cos(ij )d (49) ij π 22 0 xx++g g In a similar way as for the case of a single-wire line, an accurate logarithmic approximation is proposed by Rachidi et al. [15] which is given by

2 2 hh+ r ++ij ij 1 gggg jwm 22 ′ ≅ 0  Zg ln (50) ij 4π hh+ 22 r ij+ ij gggg 22

Note that in eqns (43) and (44), the terms corresponding to the wire impedance and the so-called ground admittance have been neglected. This approximation is valid for typical overhead power lines [9]. The boundary conditions for the two line terminations are given by

hi se=− + [VZIEzziiz (0)] [A ][ (0)]∫ (0, )d (51)  0

hi se=+ [VLiiz ( )] [ ZB ][ IL ( )]∫ ELzz ( , )d (52)  0 in which [ZA] and [ZB] are the impedance matrices at the two line terminations.

6 Time-domain representation of the coupling equations

A time domain representation of the ﬁ eld-to-transmission line coupling equations is sometimes preferable because it allows the straightforward treatment of non- linear phenomena as well as the variation in the line topology [4]. On the other hand, frequency-dependent parameters, such as the ground impedance, need to be represented using convolution integrals. The ﬁ eld-to-transmission line coupling eqns (43) and (44) can be converted into the time domain to obtain the following expressions: ∂∂ ∂ [vxtse (,)][++⊗= L′′ ] [ ixt (,)]x [ ixt (,)] [ Exht (, ,)] (53) iijigij i xi ∂∂xt ∂ t

∂∂ ++′′ss = [ixtiijiiji ( , )] [ G ][ vxt ( , )] [ C ] [ vxt ( , )] 0 (54) ∂∂xt in which ⊗ denotes convolution product and the matrix x' is called the transient [ gij ] ground resistance matrix; its elements are deﬁ ned as  Z′ − gij x′ = F 1  gij (55) jw 

The inverse Fourier transforms of the boundary conditions written, for simplicity, for resistive terminal loads read

hi =− +e [vtiiz (0, )] [ RitA ][ (0, )]∫ E (0, ztz , )d (56)  0

hi =+e [()][vLiiz RB ][(0)] i∫ E (,,)d Lztz (57)  0 where [RA] and [RB] are the matrices of the resistive loads at the two line terminals. The general expression for the ground impedance matrix terms in the frequency domain (49) does not have an analytical inverse Fourier transform. Thus, the elements of the transient ground resistance matrix in the time domain are to be, in general, determined using a numerical inverse Fourier transform algo- rithm. However, analytical expressions have been derived which have been shown to be reasonable approximations to the numerical values obtained using an inverse FFT (see Chapter 2).

7 Frequency-domain solutions

Different approaches can be employed to ﬁ nd solutions to the presented coupling equations. Sections 7 and 8 present some commonly used solution methods in the frequency domain and in the time domain, respectively. To solve the coupling equations in the frequency domain, it is convenient to use Green’s functions that relate, as a function of frequency, the individual coupling sources to the scattered or the total voltages and currents at any point along the line. Green’s functions solutions require integration over the length of the line, where the distributed sources are located. This approach is the subject of Section 7.1. Under special conditions, it is possible to obtain more compact solutions or even analytical expressions. In particular, if solutions are required at the loads only, it is possible to write the load voltages and currents in a compact manner, with the complexity essentially hidden in the source. This formulation, termed the BLT equations, will be presented in Section 7.2.

7.1 Green’s functions

The ﬁ eld-to-transmission line coupling equations, together with the boundary conditions, can be solved using Green’s functions, which represent the solutions for line current and voltage due to a point voltage and/or current source [1]. In this section, we will present the solutions, using the Agrawal et al. model (Section 2.2) for the case of a single-conductor line. Similar solutions can be found for the case of a multiconductor line (see e.g. [1, 3]).

Considering a voltage source of unit amplitude at a location xs along the line (since only distributed series voltage sources are present in the model of Agrawal et al., it is not necessary to consider a parallel unitary current source), the Green’s functions for the current and the voltage along the line read, respectively [1],

−gL e −− − − =−−gggg()xL>><< () xL x x GxxI (; )− (err e )(e e ) (58) s212(1Z − rr e2gL ) c12

−gL d e −− − − =+−gggg()xL>><< () xL x x GxxV (;s21 )− (edr e )(e dr e ) (59) 2(1− rr e2gL ) 12 where

• x< represents the smaller of x or xs, and x> represents the larger of x or xs. • d = 1 ____for x > xs and d = –1 for x < xs. √ • g = Z'Y'____ is the complex propagation constant along the transmission line, √ • Zc = Z'/Y' is the line’s characteristic impedance. • r1 and r2 are the voltage reﬂ ection coefﬁ cients at the loads of the transmission line given by ZZ−− ZZ rr==Ac, Bc (60) 12ZZ + ZZ + Ac Bc

ZA and ZB are the termination impedances as illustrated in Fig. 1. The solutions in terms of the total line current and scattered voltage can be written as the following integrals of the Green’s functions [1]

Lhh =+′ ee − Ix()∫∫∫ GxxVIIzIz (;ss ) d x s Gx (;0) E (0,)d z zGxLELzz (; ) (,)d (61) 000

Lhh see=+′ − Vx()∫∫∫ GxxVxVVzVz (;ss ) d s Gx (;0) E (0,) zdzGxLE (; ) (0,)d zz(62) 000 Note that the second and the third terms on the right-hand side of eqns (61) and (62) are due to the contribution of equivalent lumped sources at the line ends (see Fig. 3). The total voltage can be determined from the scattered voltage by adding the contribution from the exciting ﬁ eld as

h =−se Vx() V () x∫ Ez (,)d xz z (63) 0

7.2 BLT equations

If we are interested in the transmission line response at its terminal loads, the solutions can be expressed in a compact way by using the so-called BLT (Baum, Liu, Tesche) equations [1]

−1 I(0) 10−−rr eg L S = 1/Z 111 (64) c − g L IL() 01r22e −r S 2

−1 V(0) 10+−rr eg L S = 111  (65) + g L VL() 01r22e −r S 2 where the source vector is given by

Lhhg L 1 g x ee1e e ∫∫∫es Exhx ( , )d+− E (0, z )d z ELzz ( , )d  2 xzss22 z S1 000 = (66) S Lhhg L 2 −1 g()Lx− eeee1 ∫∫∫e(,)d(0,)d(,)ds Exhx−+ E zz ELzz 2 xzzss22 000

Note that in the BLT equations, the solutions are directly given for the total voltage and not for the scattered voltage. For an arbitrary excitation fi eld, the integrals in eqn (66) cannot be carried out analytically. However, for the special case of a plane wave excitation fi eld, the integrations can be performed analytically and closed-form expressions can be obtained for the load responses. General solutions for vertical and horizontal fi eld polarizations are given in [1].

8 Time-domain solutions

Several approaches can be used to solve the coupling equations in the time domain [1, 3]. We will present here simple analytical expressions that can be obtained for the case of a lossless line involving infi nite summations. In Chapter 2, the Finite Difference Time Domain (FDTD) technique is applied to obtain general solutions of fi eld-to-transmission line coupling equations including frequency-dependent losses. Under the assumption of a lossless line, it is possible to obtain analytical solutions for the transient response of a transmission line to an external fi eld excitation ω [1]. In this case, the propagation constant becomes_____ purely imaginary g = j /c and √ the characteristic impedance is purely real Zc = L'/C' . If we assume further that the termination impedances are purely resistive, the refl ection coeffi cients r1 and –2gL r2, too, become real. For | r1r2e | < 1, the denominator in Green’s functions (58) and (59) can be expanded to

∞ 1 − = ∑ (err jLcnw2/ ) (67) −2gL 12 (1− rr e ) = 12 n 0 This equation is not valid for the case of a lossless line with reﬂ ection coefﬁ cients –2gL of magnitude 1, in which the condition | r1r2e | = 1 will be met at a number of resonance frequencies causing the solutions to be unbounded.

With the above transformation, it is easy to show that all the frequency depen- dences in eqns (64) and (65) will be in the form e–jwt, t being a constant. Therefore, it is possible to convert the frequency domain solutions to the time domain analytically to obtain the following transient responses for the load voltages (for details, see [1]) ∞ +− + n 1 2(nLx 1)ss 2 nLx v(0, t )=+ (1rrrr )∑ ( )  vt − − vt − 1122s2 cc s n=0 (68) ∞ +− ++ n 1 2(nLx 1)ss 2( L 1) x vLt(,)=+ (1rrr )∑ ( )  v t − − r v t − 212s2 cc 1s n=0 (69) where

Lhh =+−eee vtsss()∫∫∫ Exzz ( xhtx , ,)d E (0,,)d ztz E ( Lztz ,,)d (70) 000 e e e Note that E x ( xs,h,t), Ez (0, z,t) and E z ( L,z,t) are the time-domain components of the exciting ﬁ eld.

9 Conclusions

We discussed the transmission line theory and its application to the problem of external electromagnetic fi eld coupling to transmission lines. After a short discussion on the underlying assumptions of the transmission line theory, the fi eld-to- transmission line coupling equations were derived for the case of a single wire line above a perfectly conducting ground. Three different but completely equivalent approaches that have been proposed to describe the electromagnetic fi eld coupling to transmission lines were also presented and discussed. The derived equations were extended to deal with the presence of losses and multiple conductors. The time-domain representation of fi eld-to-transmission line coupling equations which allows a straightforward treatment of non-linear phenomena as well as the variation in the line topology was also described. Finally, solution methods in the frequency domain and the time domain were presented.

References

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